I'm not sure if this belongs in physics stackexchange but I am curious as to why we study solvmanifolds. Of course there are instances with Einstein solvmanifolds and solvmanifolds as a subset of nilmanifold theory, but im not sure what importance solvmanifolds carry. thank you!
2026-03-27 22:01:35.1774648895
Why are solvmanifolds of importance?
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Let me just give an example, where solvmanifolds are still studied today - see the comment above. There is a famous open conjecture by L. Auslander:
Conjecture (Auslander): Every compact complete affine manifold has virtually polycyclic fundamental group - in this case, the manifold is finitely covered by an affine solvmanifold $G/\Gamma$, where $G$ is a (necessarily solvable) Lie group with a left-invariant complete affine structure and $\Gamma\subset G$ is a lattice.
Abels, Margulis and Soifer have worked more than a decade on it. However, so far, only special cases have been solved.
Equivalently, every affine crystallographic group is virtually solvable. Or even differently, the fundamental group of a compact complete affinely-flat manifold is virtually solvable.
For a survey from a geometrical viewpoint see
W. M. Goldman: Two papers which changed my life: Milnor’s seminal work on flat manifolds and bundles. Frontiers in complex dynamics, Princeton Math. Ser. 51 (2014), 679–703.